Linear resolution of products of monomial ideals related to maximal minors
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866914354252742656 |
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| author | Banerjee, Arindam Ghosh, Dipankar Selvaraja, S. |
| author_facet | Banerjee, Arindam Ghosh, Dipankar Selvaraja, S. |
| contents | Let $ X $ be an $ m \times n $ matrix of distinct indeterminates over a field $ K $, where $ m \le n $. Set the polynomial ring $ K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n] $. Let $ 1 \le k < l \le n $ be such that $ l - k + 1 \ge m $. Consider the submatrix $ Y_{kl} $ of consecutive columns of $ X $ from $ k $th column to $ l $th column. Let $ J_{kl} $ be the ideal generated by `diagonal monomials' of all $ m \times m $ submatrices of $ Y_{kl} $, where the diagonal monomial of a square matrix means product of its main diagonal entries. We show that $ J_{k_1 l_1} J_{k_2 l_2} \cdots J_{k_s l_s} $ has a linear free resolution, where $ k_1 \le k_2 \le \cdots \le k_s $ and $ l_1 \le l_2 \le \cdots \le l_s $. This result is a variation of a theorem due to Bruns and Conca. Moreover, our proof is self-contained, elementary and combinatorial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1902_09748 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Linear resolution of products of monomial ideals related to maximal minors Banerjee, Arindam Ghosh, Dipankar Selvaraja, S. Commutative Algebra Primary 13D02, 15A15 Let $ X $ be an $ m \times n $ matrix of distinct indeterminates over a field $ K $, where $ m \le n $. Set the polynomial ring $ K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n] $. Let $ 1 \le k < l \le n $ be such that $ l - k + 1 \ge m $. Consider the submatrix $ Y_{kl} $ of consecutive columns of $ X $ from $ k $th column to $ l $th column. Let $ J_{kl} $ be the ideal generated by `diagonal monomials' of all $ m \times m $ submatrices of $ Y_{kl} $, where the diagonal monomial of a square matrix means product of its main diagonal entries. We show that $ J_{k_1 l_1} J_{k_2 l_2} \cdots J_{k_s l_s} $ has a linear free resolution, where $ k_1 \le k_2 \le \cdots \le k_s $ and $ l_1 \le l_2 \le \cdots \le l_s $. This result is a variation of a theorem due to Bruns and Conca. Moreover, our proof is self-contained, elementary and combinatorial. |
| title | Linear resolution of products of monomial ideals related to maximal minors |
| topic | Commutative Algebra Primary 13D02, 15A15 |
| url | https://arxiv.org/abs/1902.09748 |